QFT as a set of ODEs

A bstract Correlation functions of local operators in Quantum Field Theory (QFT) on hyperbolic space can be fully characterized by the set of QFT data {∆ i , C ijk , $$ {b}_j^{\hat{\mathcal{O}}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>b</mml:mi> <mml:mi>j</mml:mi> <mml:mover> <mml:mi>O</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:msubsup> </mml:math> }. These are the scaling dimensions of boundary operators ∆ i , the boundary Operator Product Expansion (OPE) coefficients C ijk and the Boundary Operator Expansion (BOE) coefficients $$ {b}_j^{\hat{\mathcal{O}}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>b</mml:mi> <mml:mi>j</mml:mi> <mml:mover> <mml:mi>O</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:msubsup> </mml:math> that characterize how each bulk operator $$ \hat{\mathcal{O}} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>O</mml:mi> <mml:mo>̂</mml:mo> </mml:mover> </mml:math> can be expanded in terms of boundary operators $$ {\mathcal{O}}_j $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>O</mml:mi> <mml:mi>j</mml:mi> </mml:msub> </mml:math> . For simplicity, we focus on two dimensional QFTs and derive a universal set of first order Ordinary Differential Equations (ODEs) that encode the variation of the QFT data under an infinitesimal change of a bulk relevant coupling. In principle, our ODEs can be used to follow a Renormalization Group (RG) flow starting from a solvable QFT into a strongly coupled phase and to the flat space limit.